Raheem Sterling and Joe Allen scored in the 2-0 win at Anfield as Liverpool returned to fifth position - four points off Manchester City in the final UEFA Champions League place.
With City's form having stuttered of late, Rodgers said it was vital to ensure maximum points after back-to-back league defeats to Manchester United and Arsenal.
"Very important, we knew we had to win tonight and we'll take that into the [FA Cup semi-final at the] weekend," the Northern Irishman told Sky Sports.
"Manchester City have maybe opened the door up for those below but there's still so many twists left.
"Our attacking intent was good, we started slowly, and after the wonderful [Sterling] goal we controlled the game though to the last six or seven minutes of the first half.
"We gave the ball away too cheaply after that, which is probably the only criticism of us this evening.
"Overall, the standard of performance, after a midweek game, the players deserve huge credit."
Captain Jordan Henderson echoed his managers views, adding: "We knew it was an important game for us as the last few weeks have been disappointing performances and results for us in the league.
"We had a few chances and I thought we controlled the game for 30, 35 minutes. We went a bit sloppy after that and could have kept it better.
"But we kept going and Joe wrapped it up for us, plus a clean sheet is always pleasing."
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
